For network systems, the interconnection structures of the networks are crucial in controlling, monitoring, and so on [1]}, [2]}, [3]}, [4]}.
The model reduction methods which preserve the interconnection structure of networks have been proposed for a few decades. The network reduction method [5]} is proposed to preserve the scale-free property of networks by formulating the interconnection constraints as the eigenvector centrality. In this method, the reduced network remains a flow network if the initial network is a flow network. Furthermore, [6]} introduces network reduction methods which preserve the interconnection structure of subsystems.
Moreover, positive network systems whose outputs are always nonnegative under nonnegative inputs are important because
the systems are often found in real world applications such as
pharmacokinetics, metabolism, epidemiology, ecology, and logistics [7]}, [8]}.
In a positivity-preserving manner, some model reduction methods have been proposed in [9]}.
For instance, the clustering-based method [10]} is one of the most known methods which preserves the positivity.
Furthermore, in [10]}, the theoretical bound of the \(H^2\) -error between the original and reduced systems is provided.
Nonetheless, the clustering-based method does not guarantee the \(H^2\) -optimality between the original and reduced network systems.
